#113086

order	512
nilpotency class	3
isoclinism	isoclinic class of #10078
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂²×C₄	#120
commutator	C₂×C₄²	C₂²×C₄
Frattini, cosocle	C₂²×C₄²	C₂³
socle	C₂³	#379
prime radical	C₂×#121	D₄
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

847 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 52
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 3 generators on 52 points:

(6,10,12,15)(11,19,20,18)(21,22)(23,31,34,27)(24,35,36,46)(25,30)(26,29)(28,37,43,39)(32,48)(33,45,38,42)(40,44)(41,50)(47,51)(49,52);(1,2)(3,4)(5,6)(7,11)(8,15)(9,12)(10,17)(13,18)(14,20)(16,19)(21,23)(22,27)(24,32)(25,38)(26,34)(28,41)(29,31)(30,45)(33,40)(35,48)(36,49)(37,51)(39,50)(42,44)(43,47)(46,52);(2,4)(5,7,9,14)(6,11,12,20)(8,16,17,13)(10,18,15,19)(21,24)(22,28)(23,32)(25,39)(26,36)(27,41)(29,43)(30,46)(31,47)(33,51)(34,49)(35,44)(37,40)(38,50)(42,48)(45,52)

Polycyclic presentation:

2^9|a^2=g,c^2=i,d^2=i,e^2=h,b^a=bd,c^a=ce,d^a=dh,c^b=cf,d^b=di,e^b=ehi,g^b=ghi,d^c=dhi,e^c=eh,g^c=gh

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 20282);